Optimal. Leaf size=103 \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}-\frac {2 a \left (c+d x^2\right )^{3/2} (5 b c-a d)}{15 c^2 x^3}-\frac {b^2 \sqrt {c+d x^2}}{x}+b^2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {462, 451, 277, 217, 206} \[ -\frac {a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}-\frac {2 a \left (c+d x^2\right )^{3/2} (5 b c-a d)}{15 c^2 x^3}-\frac {b^2 \sqrt {c+d x^2}}{x}+b^2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 277
Rule 451
Rule 462
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^6} \, dx &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}+\frac {\int \frac {\left (2 a (5 b c-a d)+5 b^2 c x^2\right ) \sqrt {c+d x^2}}{x^4} \, dx}{5 c}\\ &=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}-\frac {2 a (5 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^3}+b^2 \int \frac {\sqrt {c+d x^2}}{x^2} \, dx\\ &=-\frac {b^2 \sqrt {c+d x^2}}{x}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}-\frac {2 a (5 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^3}+\left (b^2 d\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx\\ &=-\frac {b^2 \sqrt {c+d x^2}}{x}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}-\frac {2 a (5 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^3}+\left (b^2 d\right ) \operatorname {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )\\ &=-\frac {b^2 \sqrt {c+d x^2}}{x}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{5 c x^5}-\frac {2 a (5 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^3}+b^2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 104, normalized size = 1.01 \[ b^2 \sqrt {d} \log \left (\sqrt {d} \sqrt {c+d x^2}+d x\right )-\frac {\sqrt {c+d x^2} \left (a^2 \left (3 c^2+c d x^2-2 d^2 x^4\right )+10 a b c x^2 \left (c+d x^2\right )+15 b^2 c^2 x^4\right )}{15 c^2 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 221, normalized size = 2.15 \[ \left [\frac {15 \, b^{2} c^{2} \sqrt {d} x^{5} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) - 2 \, {\left ({\left (15 \, b^{2} c^{2} + 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + {\left (10 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, c^{2} x^{5}}, -\frac {15 \, b^{2} c^{2} \sqrt {-d} x^{5} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left ({\left (15 \, b^{2} c^{2} + 10 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + {\left (10 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{15 \, c^{2} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.47, size = 403, normalized size = 3.91 \[ -\frac {1}{2} \, b^{2} \sqrt {d} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right ) + \frac {2 \, {\left (15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c \sqrt {d} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b d^{\frac {3}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{2} \sqrt {d} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c d^{\frac {3}{2}} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} d^{\frac {5}{2}} + 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{3} \sqrt {d} + 40 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{2} d^{\frac {3}{2}} + 10 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{4} \sqrt {d} - 20 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{3} d^{\frac {3}{2}} + 10 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{2} d^{\frac {5}{2}} + 15 \, b^{2} c^{5} \sqrt {d} + 10 \, a b c^{4} d^{\frac {3}{2}} - 2 \, a^{2} c^{3} d^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 123, normalized size = 1.19 \[ b^{2} \sqrt {d}\, \ln \left (\sqrt {d}\, x +\sqrt {d \,x^{2}+c}\right )+\frac {\sqrt {d \,x^{2}+c}\, b^{2} d x}{c}-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} b^{2}}{c x}+\frac {2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2} d}{15 c^{2} x^{3}}-\frac {2 \left (d \,x^{2}+c \right )^{\frac {3}{2}} a b}{3 c \,x^{3}}-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}} a^{2}}{5 c \,x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 94, normalized size = 0.91 \[ b^{2} \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {\sqrt {d x^{2} + c} b^{2}}{x} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b}{3 \, c x^{3}} + \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d}{15 \, c^{2} x^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{5 \, c x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.50, size = 199, normalized size = 1.93 \[ - \frac {a^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c x^{2}} + \frac {2 a^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{15 c^{2}} - \frac {2 a b \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {2 a b d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3 c} - \frac {b^{2} \sqrt {c}}{x \sqrt {1 + \frac {d x^{2}}{c}}} + b^{2} \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {b^{2} d x}{\sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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